Self-maps of modules over
the Steenrod algebra
John H. Palmieri
April 1992
Journal ofPure and Applied Algebra 79 (1992) 281-291
Abstract
For a finite module M over the Steenrod algebra A, we prove
the existenceof a non-nilpotent element in Exts;tA(M;M) "parallel to
the vanishing line." We use this result to give a proof of Margolis'
construction to kill Pts-homology groups, at all primes.
Let A be the mod p Steenrod algebra. It is well-known (see [8],for example)
that its dual A is isomorphic (as!an algebra) to F2[1; 2; : :]:when p = 2,
and to Fp[1; 2; : :]: E[|0; |1;!:!:]:when p is odd. Define the Milnor basis
of A to be the dualsto the monomial!basis!of A ;let Ptsbe the Milnor basis
element dual to pt, and, when p!is odd,let Qt be dual to |t. It is easy to
!
check that when s < t then (Pst)p!=0, and for p odd, (Qt)2 =0 for all t.
!
Therefore we call these elements!differentials, and given any differential x
and any A-module M we can define!the!homology!of!M with respect to x
by: !!!!
!!ker!Pst:!M ! M
H(M ;Pst) = ;
im!(Pst)p1!: M ! M
and for p odd !!
k!erQt!: M !M
H (M;Qt) = :
im Qt: M ! M
Given a differential x, define its slope, sl ope(x), by
!
pjPtsj!!
slope(Pst) = ;
2
and
slope(Qt) = jQtj:
(This notation is motivated by Theorem 1.4.) The differentials are linearly
ordered by slope; say that a module M is type hm;ni (also written M =
Mhm; ni) if and only if H(M; x) = 0 whenever slope(x) < m or slope(x) > n.
We prove the following theorem.
Theorem A Fix a differential x with slope(x) = m. Given a finite module
M = Mhm; 1i with H(M; x) 6= 0, there is an element v 2Extk;kmA(M; M)
for some k which is non-nilpotent under Yoneda composition.
This is a Steenrod algebra analog of the theorem of Hopkins and Smith that
any finite p-local spectrum X with K(n 1) (X) = 0 and K(n) (X) 6= 0has
a vn-map, a non-nilpotent self-map that induces an isomorphism on K (n)-
homology (see [4]). (In the appropriate setting, the map v in Theorem A
induces an isomorphism on x-homology.) Theorem A is a generalization of a
result used by Hopkins and Smith to prove the theorem for spectra. We ac-
tually prove Theorem A for a slightly larger collection of modules than finite
ones, namely for "stably finite" modules (see Section 1). In Section 3, for
each differential x we construct a stably finite module M = Mhslope(x);1i
with H(M; x) 6= 0,solving the algebraic analog of the problem of construct-
ing a finite spectrum with a vn-map. Note that for each x one can also
construct a finite module M of this sort,by taking a tensor product of "pth
powers" of sub-Hopf algebras A(n) of A for particular n's, using Mitchell's
A-module structure. At the prime 2, for example, the module A(1) A(1)
(where is the doubling functor) is a module of type h7;1i. See [10].
Using Theorem A, we give a simple proof of
Theorem B ([6] for p = 2) Given any boundedbelow module M and any
integer m 1,
(a) thereexists a bounded below module N= N hm;1i and a map f : M !
N so that H(f;x) is an isomorphism if slope(x) m;